Magnetic thin films have complicated domain patterns that may or may not repeat with each cycle through a hysteresis loop. A magnetic thin film with perpendicular anisotropy, such as that used in computer hard drives, for example, commonly exhibits labyrinthine domain patterns. These patterns are disordered over a macroscopic length scale, and intuitively we do not expect to observe any symmetry in such systems. Scientists at the ALS, the University of Oregon, and the University of California, San Diego, have recently used coherent soft x-ray scattering with angular Fourier analysis to discover that the disordered domain patterns do, in fact, exhibit rotational symmetries, which can be as small as two-fold or as large as 30-fold. Their study of magnetic symmetries gives scientists a toolbox for discovering hidden symmetries in diverse material systems.

The Influence of Unseen Symmetries

One of the most powerful tools in the mathematics of science, from the physics of elementary particles to macroscopic matter, is symmetry, no doubt reflecting the ubiquitous occurrence of symmetry in nature. The easiest symmetries to visualize are structural—for example, the six-fold rotationally periodic structures of snowflakes. The translational periodicity represented by the repeated pattern that makes up the structure of a crystalline material is another example. Structure is not the only physical property that exhibits symmetry in nature; some of these properties are quite far removed from everyday experience, such as quantum properties like the spin of elementary particles.

As it happens, in many cases, the existence of a symmetry is not immediately apparent, and it is called a hidden symmetry. Nonetheless, there may be clues. Consider the case where a system (general name for simple or complex samples of all kinds) is microscopically disordered but nevertheless exhibits a patterned structure on a longer mesoscopic length scale. One way for this pattern to emerge from building blocks that are nominally random and disordered is through hidden symmetries that exist over several length scales and that conspire to produce the large-scale pattern. Research reported by Su et al. shows how to address such esoteric questions, while specifically suggesting that hidden rotational symmetries may play an important role in determining the overall structure of a nominally disordered system.

The occurrence of symmetry is ubiquitous in nature. Common symmetries are easy to detect, for example, in geometric structures and periodic patterns. However, a diverse array of physical systems, ranging from quarks to quasicrystals, possess underlying symmetries that are hidden and must be discovered by combining discerning experimental techniques with detailed analysis. Particularly interesting are systems that are microscopically disordered but nevertheless exhibit a pattern on a longer mesoscopic length scale. How does this pattern emerge when the building blocks are nominally random and disordered? Is it possible that the system has hidden symmetries that exist over several length scales and that conspire to produce the large-scale pattern? A similar question can be posed for a structural glass, where local high-symmetry structures probably play an important role in the glass transition. The research highlighted here paves a path for understanding such esoteric questions and suggests that, indeed, hidden symmetries may play an important role in determining the overall structure of a nominally disordered system.

To reveal these symmetries, the researchers used coherent soft x-ray scattering with the photon energy tuned to be resonant with the Co L3 edge, thereby providing the magnetic contrast required to visualize magnetic domains. The coherent scattering endstation at Beamline 12.0.2 uniquely combines the ability to provide in-situ magnetic fields up to 0.6 T in any arbitrary direction, low-temperature control, and high coherent x-ray flux. The sample used was a Co/Pd multilayer film that shows perpendicular anisotropy, meaning the magnetization points out of the plane of the thin film sample. Such a perpendicular anisotropy system scatters coherent x-rays into a cone modulated with "magnetic speckles." Angular correlation analysis of narrow annular regions in such speckle patterns revealed a "treasure" of symmetries whose emergence depended on various factors, including temperature, scattering wave vector, applied magnetic field, and magnetization history.

Left: A typical speckle pattern from the Co/Pd multilayer. Color bar at bottom indicates relative intensity. The rotational symmetry of a scattering pattern is detected by dividing it up into annular slices (shown in red) corresponding to a fixed magnitude of the scattering vector, and calculating the angular autocorrelation function. Right: Examples of angular autocorrelation functions where the rotational symmetries are readily apparent. For each length scale and applied field, a variety of symmetries can occur.

For example, one map of the spectrum of observed symmetries as a function of the applied field normal to the plane of the sample at a scattering wave vector of 0.264 nm-1, corresponding to a length scale of 237 nm, reveals that there are "symmetry hot spots" with rotational order equal to 4, 14, 16, and 24 as one traverses the lower half of the hysteresis loop. These symmetries tend to appear and disappear at fields at which large Barkhausen cascades (noise in the change in the magnetization of a ferromagnet when the applied field is changed) are expected, suggesting a connection between hidden symmetries and magnetization reversal.

Map of the spectrum of hidden rotational symmetries as a function of applied magnetic field at a particular scattering wave vector (annular radius in the first figure). The symmetries are driven in and out of existence preferentially in places on the magnetization loop where large Barkhausen events are expected.

The rotational symmetries studied are so localized in wave-vector (q) space that they would be difficult to detect with real-space probes directly. Seeking such hidden symmetries in Fourier space and measuring their evolution with temperature, time, and applied field may provide a way to understand the emergent mesoscale pattern formation in a diverse array of complex materials, including labyrinthine domain patterns observed in many other soft, hard, and biological systems having uniaxial anisotropy.